Page:Great Neapolitan Earthquake of 1857.djvu/180

height and length of the parallelopiped, overturning on $\gamma$.

Let the thickness or third dimension of this rectangular plate be $\tau$, and let it be supposed applied to that side of the parallelopiped, to which the wedges were adherent, and added to the thickness $t$, or width of base of the parallelopiped, for the value of $\beta =t+\tau$. Further, $h$ being the height of the end wall or parallelopiped, let its altitude be assumed increased in the proportion

$t:h::t+{\frac {1}{4}}\tau :\alpha$

for the new value of $\alpha$. The case now resolves itself into that of Eqs. V. and VI., substituting in these the new values of $\alpha$ and $\beta$, thus obtained; in any case worth practical application this may be done without sensible error.

Case 7th. Angular wedge thrown over upon its apex.

This is the case referred to pp. 66-72, in treating of fissures, as one of frequent occurrence, and valuable in deciding direction of wave-path. It can, however, be very nearly applied to the determination of velocity. The problem, generally treated, leads to very complex results; and approximations are equally tedious, except in the case in which the direction of the wave-path is parallel to one of the external sides of the wedge, when the wedge vanishes and the case becomes identical with the last one.

What has preceded refers only to horizontal force or velocity $V$ (normal wave). We now proceed to